It is a bit more complicated than that.
The main abdominal muscles are the rectus abdominis, external and internal obliques, and the transverse abdominis. The purpose of these muscles is two-fold: on the one hand, they move the spine (bending forward and twisting), on the other hand, they are the muscles used for forced exhalation.
These two functions are pronounced differently, depending on how deep the muscle is. The main function of the more superficial muscles (the rectus abdominis and the external obliques) is bending and twisting, and the main function of the deepest muscles (mainly the transverse abdominis) is compression and forced exhalation.
Given this complexity, it is not surprising, that when people say they are "contracting the abs", they can mean different things. Some people mean as if they are trying to do a crunch (bending the spine), some mean exhalation. Of course, most people always do a bit of both, as it is not easy to isolate the abdominal muscles (though it is definitely possible to isolate them, and it looks super weird). But people can be using different muscles slightly differently, according to their natural disposition.
When lifting heavy objects, like squatting with weight, it is natural to hold the breath. As described in Christian's answer, the Valsalva maneuver is forced exhalation without letting the air out. It increases abdominal pressure, like an inflated tire, it makes the torso more rigid and stable. Many people do this naturally without being told. The main muscle responsible for increasing this intra-abdominal pressure will be the transverse abdominis:
... the co-ordinative patterns shown between the muscles of the ventrolateral abdominal wall are task specific based upon demands of movement, torque and stabilization. It appears that transversus abdominis is the abdominal muscle whose activity is most consistently related to changes in intra-abdominal pressure. 
It is normal, that, when the intra-abdominal pressure increases, the abdomen is pushed out a little bit. The stomach being pushed out a little bit does not mean, that the transverse abdominis is inactive, it is still working to balance out the pressure, which helps a lot with squatting.
I suspect, that when you say you are contracting your belly consciously, you are actually contracting the more superficial muscles, like when doing a crunch. They do not help much with squatting, in fact, they can make it harder, because they work to bend the spine forward, so you are working against yourself in the squat. Also, when you say you are doing front squats with "belly relaxed", you are probably contracting the deeper muscles, you are just less conscious of it.